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The Z-score is a very useful measure in statistics that tells us how far and in what direction a data point deviates from the mean of a data set. It's calculated by taking the difference between a value and the mean, and then dividing by the standard deviation. For any normal distribution:

• If the Z-score is 0, the value is exactly at the mean.
• A positive Z-score indicates the value is above the mean.
• A negative Z-score signifies the value is below the mean.

In our exercise, the Z-score helps determine how unusual it is for a hopper car's load to be less than 74.5 tons when the mean is 75 tons. Using the formula for Z-score: \[ Z = \frac{X - \mu}{\sigma} \] we calculate for a single car: \[ Z = \frac{74.5 - 75}{0.8} = -0.625 \] This negative Z-score shows the weight is below the mean. To find the probability of this occurrence, we refer to the standard normal distribution tables.

The Sample Mean is the average value in a sample set, and it's a key concept in statistics for estimating the attributes of a population. When considering a group of 20 hopper cars and their coal weights, you need to look at how their average (sample mean) compares to the expected average (population mean). The formula to calculate the sample mean is the sum of all measurements divided by the number of samples. To determine how likely it is for the average load of 20 cars to be less than 74.5 tons, a Z-score for the sample mean is calculated differently. You use the standard error instead of the standard deviation alone: \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \] This represents the standard deviation of the sampling distribution of the sample mean. For our exercise: \[ \sigma_{\bar{x}} = \frac{0.8}{\sqrt{20}} \approx 0.179 \] Knowing this allows us to compute: \[ Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} = \frac{74.5 - 75}{0.179} \approx -2.7933 \] A Z-score of -2.7933 indicates the sample mean is significantly below the population mean.

Cumulative Probability helps to determine the likelihood that a random variable will fall within a certain range. It sums up the probabilities of all values less than or equal to a certain number. For Z-scores, the cumulative probability is often used to assess how extreme a calculated Z-score is. In our case, if you have a Z-score of -0.625 for a single car's load, you'll look up this score in a standard normal distribution table to find the proportion of data falling below this Z-score. The cumulative probability associated with a Z-score of -0.625 is approximately 0.2659. This means there's a 26.59% chance that a single car carries less than 74.5 tons. When considering multiple cars, the cumulative probability for a Z-score of -2.7933 is approximately 0.0026. This is significantly lower, indicating only a 0.26% chance that the average weight of 20 cars is under 74.5 tons. Such a low cumulative probability suggests it is quite unusual, possibly pointing to an error or issue with the loader if observed in real situations.